Divergence

1. Integral theorems
Applications:

2. The Divergence
Theorem

3. Divergence of a vector Field
The divergence of a vector field are a point is a scalar
quantity of magnitude equal to flux of that vector field
diverging out per unit volume through that point in
mathematical from, the dot product of del operator ∇ and
the vector field A(x,y,z,) gives the divergence of a vector
field A. i.e. ,
div A. = ∇.A

4. But ∇ = Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z and A = ÎAX + ĵ AY + AZ k
∇. A = (Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z).(ÎAX + ĵ AY + AZ k)
∇. A = ( ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)
divA = ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z

5. Gauss Divergence Theorem
According to this theorem the volume intrigle of
divergence of a vector field A over a volume V is equal
to the surface integral of that vector field A taken over
the surface S which encloses that volume V. I ,. e.
∫∫∫(divA)dV = ∫∫A. da
Thus this theorem is used to convert the volume
integral into the surface integral or to convert surface
integral into the volume intrgral.

6. Proof : In cortesian coordinates,
div A =∇ .A = (Î ∂∕∂x + ĵ ∂∕∂y + k∂∕∂z).(ÎAX + ĵ AY + AZ k)
= ( ∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)
And dV = dxdydz
While A.da = (ÎAX + ĵ AY + AZ k).(Îdax + ĵday + kdaz)
= Axdax + Ayday + Azdaz
= Axdax + Ayday + Azdxdy

7. According to Gauss Theorem….
∫∫∫ (∂AX∕∂x + ∂ AY ∕∂y + ∂ AZ ∕∂z)dxdydz
=∫∫ (Axdydz + Aydxdz + Azdxdy)

8. EXAMPLE:1
Let Q be the region bounded by the sphere x 2 + y 2 + z
2 = 4. Find the outward flux of the vector field
F(x, y, z) = 2x 3 i + 2y 3 j + 2z 3k through the sphere.
Solution:

10. GREEN’S
THEOREM

11. GREEN’S THEOREM
Green’s theorem is mainly used for the
integration of the line combined with a curved
plane. This theorem shows the relationship
between a line integral and a surface integral.
It is related to many theorems such as Gauss
theorem, Stokes theorem

12. What is Green’s Theorem?
 Green’s theorem is one of the four fundamental
theorems of calculus, in which all of four are closely
related to each other. Once you learn about the concept
of the line integral and surface integral, you will come to
know how Stokes theorem is based on the principle of
linking the macroscopic and microscopic circulations.
Similarly, Green’s theorem defines the relationship
between the macroscopic circulation of curve C and the
sum of the microscopic circulation that is inside the curve
C.

14. Green’s Theorem Statement
 Let C be the positively oriented, smooth, and simple
closed curve in a plane, and D be the region
bounded by the C. If L and M are the functions of (x,
y) defined on the open region, containing D and have
continuous partial derivatives, then the Green’s
theorem
Where the path integral is traversed counterclockwise
along with C.

15. Green’s Theorem Proof
The proof of Green’s theorem is given
here. As per the statement, L and M are
the functions of (x, y) defined on the
open region, containing D and having
continuous partial derivatives. So based
on this we need to prove:

19. Green’s Theorem Example
Let us solve an example based on Green’s
theorem.

20. Green’s Theorem Applications
 Green’s Theorem is the particular case of Stokes
Theorem in which the surface lies entirely in the
plane. But with simpler forms. Particularly in a vector
field in the plane. Also, it is used to calculate the
area; the tangent vector to the boundary is rotated
90° in a clockwise direction to become the outward-
pointing normal vector to derive Green’s Theorem’s
divergence form.

21. Stoke’s
THEOREM

22. Statement:
Let S be an open surface bounded by a closed curve C and vector F be any vector point
function having continuous first order partial derivatives. Then
where ň= unit normal vector at any point of S drawn in the sense in which a right
handed screw would advance when rotated in the sense of the description of C.
Stokes’s Theorem relates a surface integral over
a surface S to a line integral around the boundary
curve of S (a space curve).

23. Example 1:
Evaluate
∫c F.dr
where:
⚫F(x, y, z) = –y2 i + x j + z2 k
⚫C is the curve of intersection of the plane y + z = 2 and the cylinder x2 + y2 =
1. (Orient C to be counterclockwise
when viewed from above.)

24. The curve C (an ellipse) is shown here

25. We first compute :

26. The elliptical region S in the plane y + z = 2 that
is bounded by C. If we orient S upward, C has
the induced positive orientation.

27. The projection D of S on the xy-plane is
the disk x2 + y2 ≤ 1

29. Example 2:
Use Stokes’ Theorem to compute ∫∫ curl F.ds
where:
⚫F(x, y, z) = xz i + yz j + xy k
⚫S is the part of the sphere
x2 + y2 + z2 = 4
that lies inside the cylinder
x2 + y2 =1
and above the xy-plane.

30. To find the boundary curve C, we solve:
x2 + y2 + z2 = 4 and x2 + y2 = 1:
 Subtracting, we get z2 = 3.
 So, z=√3 (since z > 0).

31. So, C is the circle given by:x2 + y2 = 1,
z=√3

32. A vector equation of C is:
r(t) = cos t i + sin t j + √3k 0 ≤ t ≤ 2π
Therefore, r’(t) = –sin t i + cos t j
Also we have:

33. Thus,byStokes’Theorem:

34. +h{an}k ¥0u
Presented by:
Vyshali H N
Nithin M
Hamsaveni M
Madhan P S
Rentala V M Sreeja
Harshith S